Double Angle Formula Calculator

Use our handy Double Angle Formula calculator to find the Sin2θ, Cos2θ & Tan2θ of any given angle.

What are double angle formulae?

Trigonometric functions can be written as double-angle formulas that can be expanded to multiple-angle functions such as triple, quadruple, quintuple, and so on by using the angle sum formulas, and then reapplying the double-angle formulas.

Common Trigonometric Formulae

1. sin ( A + B ) = sin A cos B + cos A sin B
2. sin ( A – B ) = sin A cos B – cos A sin B
3. cos ( A + B ) = cos A cos B – sin A sin B
4. cos ( A – B ) = cos A cos B + sin A sin B
5. tan ( A + B ) = $\frac{(tan A + tan B)}{(1 – tan A tan B)}$
6. tan ( A – B ) = $\frac{(tan A – tan B)}{(1 + tan A tan B)}$
7. sin ( A + B ) sin ( A – B ) = sin² A  – sin² B = cos² B – cos² A
8. cos ( A + B ) cos ( A  – B ) = cos² A  – sin² B = cos² B – sin² A
9. sin ( A + B + C ) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C – sin A sin B sin C
10. cos ( A + B + C ) = cos A cos B cos C – cos A sin B sin C – sin A cos B sin C – sin A sin B cos C
11. tan ( A  + B + C ) = $\frac{tan A+tan B+tan C-tan A tan B tan C}{1-tan A tan b-tan B tan C-tan C tan A}$

Some other important trigonometric formula are –

Complementary trigonometry identities

1. sin ( 90^o – θ ) = cos θ
2. cos ( 90^o – θ ) = sin θ
3. tan ( 90^o – θ ) = cot θ
4. cot ( 90^o – θ ) = tan θ
5. sec ( 90^o – θ ) = cosec θ
6. cosec ( 90^o – θ ) = sec θ

Supplementary Trigonometry Identities

1. sin (180°- θ) = sin θ
2. cos (180°- θ) = – cos θ
3. cosec (180°- θ) = cosec θ
4. sec (180°- θ)= – sec θ
5. tan (180°- θ) = – tan θ
6. cot (180°- θ) = – cot θ

Let us take an example.

Suppose we want to find the value of $\frac{cos⁡37^o}{sin⁡53^o}$

We have,     $\frac{cos⁡37^o}{sin⁡53^o} = \frac{cos⁡(90^o-53^o)}{sin⁡53^o}  = \frac{sin⁡53^o}{sin⁡53^o}$ = 1

Hence, $\frac{cos⁡37^o}{sin⁡53^o}$ = 1

Trigonometric Identities as values of functions at 2x in terms of vales at x

Now we shall introduce identities expressing the trigonometric functions as multiples of x, i.e. 2x, 3s, 4x etc. in terms of the values of x. some of the commonly used identities are –

1. sin 2x = 2 sin x cos x
2. cos 2x = cos² x – sin² x
3. cos 2x = 2 cos² x – 1  or 1 + cos 2x = 1 + 2 cos² x
4. cos 2x = 1 – 2 sin² x or 1 – cos 2x = 2 sin² x
5. tan 2x = $\frac{2 tan x}{1- tan^2 x}$
6. sin 2x = $\frac{2 tan x}{1- tan^2 x}$
7. cos 2x = $\frac{1- tan^2 x}{1+ tan^2 x}$

Formulae to Transform the Product into Sum or Difference

We have just learnt the formulae involving the identities, sin ( A  + B ), sin ( A – B ) and so on. Now we shall discuss about the identities that help convert the product of two sines or two cosines or one sine and one cosine into the sum or difference of two sines or two cosines.

These identities are –

1. 2 sin A cos B =  sin ( A  + B ) + sin ( A  – B )
2. 2 cos A cos B = cos ( A  + B ) + cos ( A  –  B )
3. 2 cos A sin B = sin ( A + B ) – sin ( A  – B )
4. 2 sin A sin B = cos ( A – B ) – cos ( A + B )

Similarly, if we substitute A + B = C and A – B = D in the above formulae, we will get,

A = $\frac{C+D}{2}$ and B = $\frac{C-D}{2}$

Therefore, new identities can be derived as –

1. sin C + sin D = 2 sin ( $\frac{C+D}{2}$ ) cos ( $\frac{C-D}{2}$ )
2. sin C – sin D = 2 sin ( $\frac{C-D}{2}$  ) cos ( $\frac{C+D}{2}$  )
3. cos C + cos D = 2 cos (  $\frac{C+D}{2}$ ) cos ( $\frac{C-D}{2}$ )
4. cos D – cos C =  2 sin ( $\frac{C+D}{2}$ ) cos ( $\frac{C-D}{2}$ )

How to use the double angle formula calculator?

 The double angle formula calculator can be used to find the double angle value of any of sin , cos  and tan .

Let us consider an example. Suppose we wish to find the values of the sin 120^o, cos 120^o and tan 120^o. How would we do that?

If we would have to go by the formula, we would write sin (180°- θ) = sin θ which would be equal to sin (180°- 60) = sin 60. Next, we would have to check for the value of sin 60^o. However, using the double angle calculator, we just enter the value of the angle and we would get our result. The following steps would be required for finding the value of sin , cos  and tan  –

Step 1 – In the first step, we need to enter the angle for the double of which we need to find the values of sin , cos  and tan . For instance, if we wish to find the value of sin 120^o, we need to enter 60 as the input value. Below is the snapshot of how the value will be entered –

Step 2 – The next step is to click on the calculate button. As soon we will click on the calculate button we can see the results on the right-hand side of the value that we had entered showing the values of sin , cos  and tan . In this way, we can find any value of sin , cos  and tan  using the double angle value calculator.